

where \(M_s^{(nlm)}\)  and \(J_s^{(nlm)}\)  are the mass and angular momentum of the scalar condensate with indices \(\lbrace n,l,m\rbrace \) , respectively. The \(\dot{E}_\mathrm {GW}^{(nlm)}\)  is the GW emission luminosity from the \(\lbrace n,l,m\rbrace \)  mode. Note that the interference between different modes is not considered, which includes transition as well as the annihilation of two scalars in different modes. The GW emitted by the former is small [1]}, [2]}, while the latter could cause \(\sim 10\%\)  variation of GW luminosity and beat-like waveforms, which is the topic of Sec. . An approximation of \(\dot{E}_\mathrm {GW}^{(011)}\)  in the limit \(M\mu \ll 1\)  is obtained in Ref. [3]} with a Schwarzschild background metric. Following the same method, we obtain the results for the other three modes considered in this work. The results are listed below,
\(\dot{E}^{(011)}_{\mathrm {GW}}&=\frac{484+9 \pi ^{2}}{23040}\Big (\frac{M_{s}^{(011)}}{M}\Big )^2(M \mu )^{14},\\\dot{E}^{(111)}_{\mathrm {GW}}&=\frac{128\left(484+9 \pi ^{2}\right)}{23914845}\Big (\frac{M_{s}^{(111)}}{M}\Big )^2(M \mu )^{14},\\\dot{E}^{(022)}_{\mathrm {GW}}&=\frac{\left.1024+49 \pi ^2\right. }{5423886846}\Big (\frac{M_{s}^{(022)}}{M}\Big )^2(M \mu )^{18} ,\\\dot{E}^{(122)}_{\mathrm {GW}}&=\frac{\left.1024+49 \pi ^2\right. }{15032385536}\Big (\frac{M_{s}^{(122)}}{M}\Big )^2(M \mu )^{18} .\) 
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